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Guided Subdivision

K. Karciauskas and J. Peters

June 29, 2005

Curvature continuous surfaces with subdivision structure are constructed
by higher-order sampling of a piecewise polynomial guide surface, at posi-
tions defined and derivatives weighted by a special, scalable reparametriza-
tion. Two variants are developed. One variant applies to the conventional
sprocket subdivision layout, say of Catmull-Clark subdivision, i.e. nested
rings consisting of N copies of L-shaped segments with three patches. The
curvature continuous surfaces are of degree (6,6). A second variant, called
polar guided subdivision, is particularly suitable for high valences N and to
cap cylindrical structures. It yields curvature continuous surfaces of degree
(4,3). Additionally, we discuss a scheme that samples with increasing den-
sity to generate a C2 surface of piecewise degree (3,3). Curvature continuity
is verified by showing convergence of anchored osculation paraboloids.

1 Introduction

Guided subdivision is a stationary subdivision procedure capable of generating,
in principle arbitrarily smooth, surfaces by sampling the composition of a guide
surface with a scalable reparametrization. While the guide surface captures the
shape implied by pre-existing data, the reparametrization is crucial to orient and
scale higher-order derivatives taken off the guide surface. While just about any
guide surface representation works, e.g. rational or the polynomial constructions
of [Pra97, Rei98], we choose a piecewise polynomial guide to get good shape
reproduction at low degree. In contrast to [Pra97, Rei98], the composition of
guide and reparametrization is not used directly but sampled. This allows us to
use the economy of spline constructions to trade more pieces for reduced poly-
nomial degree. Applying several steps of guided subdivision before capping with

Figure 1: Gauss-curvature-colored subdivision surface ring consisting of 18 seg-
ments for (left) sprocket (Ger. Zahnkranz) layout of Catmull-Clark subdivision,
(right) polar layout.

polynomial pieces is an important ingredient in the construction of high-quality
multi-sided surface blends, explained in the sequel [KPOx] to this paper, since
guided subdivision surface rings introduce fewer shape and curvature artifacts
than an equal number of conventional subdivision rings depending only on the
input mesh without guide. The publication [KPOx] also explains how guide sur-
faces are derived from boundary data. In the present paper, the guide surface is
assumed to be given so that only its structure needs to be explained.
A key new contribution of this paper is the introduction of polar guided sub-
division. Polar subdivision has a different patch layout than the conventional
sprocket-shaped arrangement of N L-shaped segments (Figure 1). While the ba-
sic structure of subdivision is the same in both layouts, the quadrilateral domain
pieces, from which the underlying topological space is built, are glued together
differently. This polar layout is particularly suitable for modeling high valence
configurations and allows for polynomial subdivision surfaces of lower degree,
namely (4,3), to achieve curvature continuity.
At first sight, guided subdivision appears to be more complex than subdivision
based on the control mesh alone. However, guided subdivision is a stationary
procedure that matches the structure of subdivision as laid out e.g. in [RP05].
Guided subdivision is also a stable procedure. Due to the guide surface, the proof
of smoothness at the extraordinary point, i.e. the limit point where N / 4 surface
segment meet, is simpler than for conventional subdivision. Our proof is based
on showing convergence of a sequence of anchored osculating paraboloids (local
quadratic functions over the tangent plane). This allows us to quantify deviation
from curvature continuity when using constructions of degree lower than (6,6) in

the sprocket layout, respectively (4,3) in the polar layout.
Guided subdivision is explained in the following steps.
Section 2 defines the layout of the guided subdivision patches (sprocket or polar).
Section 3 defines scalable reparametrizations p.
Section 4 defines the C2 guide spline g whose free parameters can be set to match
boundary data.
Section 5 defines the patches x' := H(g, o p') as Hermite interpolant of the
composition of the nth piece of the guide surface g with the nth piece of the m-
times refined reparametrization p. This defines the guided subdivision.
Section 6 assembles the pieces x' into surface rings xm; a union of rings forms a
C2 (G2) guided patchwork.
Section 7 analyzes the infinite union of surface rings, i.e. it characterizes guided
Section 8 provides curvature estimates for guided subdivision of lower degree
than is needed for curvature continuity, with the help of anchored osculating
Section 9 discusses a modification of guided subdivision, the accelerated bicubic
guided C2 scheme.
Section 10 discusses rational guided subdivision.

2 The structure of guided subdivision surfaces

Locally, a subdivision surface x E Rd is the union of a sequence of nested surface
rings x" converging to an extraordinary or centralpoint x":

x U Xm U xC .
Each surface ring x" in turn is the union of N segments x', n 0, ..., N 1 (a
segment x' can consist of several Bezier patches):
xm U xu .
The domain So of each ring x" is obtained from N copies of a basic domain
E C R2 (see Figures 2(b), 3(b)) by setting edges of (E, n) and (E, n + 1) equal
(see Figures 2(c), 3(c)):

S : x ZN, ZN := Z mod N.

Figure 2: (a) sprocket layout; (b) basic domain Yo ; (c) gluing copies of basic
domain to form S.

The rings xm of guided subdivision are special in that they are constructed by sam-
pling position and derivatives of a guide surface g with respect to a parametriza-
tion pm that, itself, has subdivision structure (as a map into Rd, d 2). That is, to
define x", we Hermite-sample g o pm where

pm : So C R2, (s,t,n) (, v),

g: 1R3, (u, v) v (x, y, z).

We distinguish two cases of reparametrization that differ in their patch layout.
sprocket layout: The basic domain is (Figure 2 (b))

o := [0,2]2 \ [0, 1)2.

Edges of copies of Yo are set equal according to

(0, s, n) = (s, 0, n + 1), s E [1..2], n E ZN
The consecutive rings xm yield the layout of Figure 2 (a), familiar from the char-
acteristic map of Catmull-Clark subdivision. In the sprocket layout, segments xm
will consist of patches of degree (k, k), i.e. degree k in each variable.
polar layout: The basic domain is (Figure 3 (b))

E := [0..1]2

O ,n zO n+1



o0 Xn

(b) 1 s (a)

Figure 3: (a) polar layout; (b) basic domain E ; (c) gluing copies of basic domain
to form SO.

Edges of copies of Eo are set equal according to

(1, t, n)= (0, t,n+ ), tE [0..1], ne ZN.
The consecutive rings x" yield the layout of Figure 3 (a). In this polar layout,
segments x" consist of patches of degree (degree circular, degree radial); we
parametrize the circular direction by s and the radial direction by t.

3 Scalable reparametrizations

In this section, we describe two maps p and p that alternatively define the coordi-
nates in which the guide surface g is Hermite sampled. They are associated with
different patch layouts
sprocket layout: The reparametrization p is associated with multi-sided gaps
in standard spline constructions. It consists of scaled copies of the characteristic
map po of Catmull-Clark subdivision (see e.g. [PR98]); po maps N copies of the
L-shaped domain Eo in Figure 2 (b) to an annulus or ring in R2. If we fix scaling
and rotation, p is uniquely determined. The reparametrization po is C2 and scaling
it by

: (c ++ 5 /(c + 9)(c + 1))/16, c: cos a, a : 2w/N,

(the subdominant eigenvalue of the Catmull-Clark subdivision matrix) results in
a copy p1 that joins po C2 in the same fashion as do rings of a Catmull-Clark

N -3 N =5 N 8 N 10

Figure 4: Alternative bicubic C2 reparametrization scalable by 1/2.

subdivision surface. (Alternatively, a good bicubic reparametrizations scalable by
1/2 for sprocket layout can be constructed if we split each bicubic into four as
shown in Figure 4. However, for the constructions in this paper, we will not need
this more complex reparametrization.)
polar layout: Next, we define the C1 polar reparametrization p of degree (2,1)
which is G2 due to symmetry. The Bezier coefficients p%, i 0..2, j 0, 1 of
one segment of a ring are shown in Figure 5. For a := 27/N,

0 = (,) 0 (1, tan(o/2)), (cos a, sin a) 0

The coefficients of segment n are defined by rotating by na about the origin.
Unlike p, there is no unique choice of scaling factor A to obtain the next inner,
adjacent ring "'+1 from p". To see this, consider the singular map psi'9 of degree
(2, 1) with coefficients p" -~n pi and p"" (0, 0), i 0, 1,2. The rings "I
are nested pieces of pS""9 since

~(s, t) psing(s, (1 Am)( t) + (1 Xm+l)t).
The simplest choice, A = 1/2, yields equicontraction but results in radially elon-
gated segments. The choice
1 + 2 sin(a/2)

creates circular and radial boundaries of roughly equal length.
Since the maps p are the parts of one map psi"', adjacent rings p and pm+1
join C" after rescaling. Simple direct verification proves also smoothness in the
circular direction as summarized in the following lemma.

(0, 0)

Figure 5: One segment of the G2 polar reparametrization p of degree (2, 1).

Lemma 1 The segments of p" are G2 connected:

S1(0, t) -= p(1, t) + k2 (1, t), k = 2 cos a 2.

4 C2 guide spline

This section explains the structure of a C2 spline of total degree d, consisting of N
C2-joined polynomial pieces in total degree Bezier form each piece defined over
a unit triangle A:
g : Q Ax ZN c R2 R3.
The degree d will be chosen so that the construction is sufficiently flexible to
well-approximate boundary data up to the second derivative. [KPOx] will explain
in more detail how to set the free parameters of this guide surface g to best match
pre-existing C2 boundary data. The construction is similar to [Pet02]. Piecewise
polynomial representation yields more flexibility at a lower degree than the con-
structions in [Pra97, Rei98, YZ04].
The rays from the center 0 := (0, 0) through the vertices

v, := (cos na, sin na), n 0, ,...,N- 1,

subdivide a plane into N sectors vOv,+l. In each sector, g is represented in total
degree, triangular Bezier form with coefficients gik, i + j + k d, labelled as
shown in Figure 6. If the boundary curves of the polynomial pieces match, the
well-known conditions for a C1 join (1) and for a C2 join (2) are (a := 27/N,
C := cosc ):

id-i-,1 --i,d-i-i + 2cgo,d-i + 2(1 c)g+,0,d-i-1; (1)
n+1 n C2 -l
i,d-i-2,2 i,2,d-i-2 4 ,d-i-1 + 4 0,d-i 4 ,,--
+ 8c(1 c)g+,o,-i-1 + 4(1 c)2 2,0,d-i-2

g- 1

Figure 6: Structure of the piecewise degree d guide spline g that defines the shape
near the extraordinary point.

By elementary computations, C2 smoothness implies the following.
(a) The six coefficients nearest to the extraordinary point, g0oo, g'-1,1,o, 8-1,o,1,
gd-2,2,0, gd2,1,1 9d-2,0,2, all have to match a single quadratic polynomial.
(in degree-raised form) defined, without loss of generality, by the six coef-
ficients with n = 0.

(b) The coefficients gd 3,2,1 satisfy the circulant system
n--1 z 2Rn, (3)
d3,2,1 + 4cg3,2,1 + gd,2, 2 (3)
Rn := 2c2g-3,3,0 + cg-3,0,3 + 4c(1 c)_2,2,0 2(1 c)gd2,1,1
+ (1 )gd-2,0,2 + 2(1 c)2 1,1,0

(c) If the coefficients g9 3,2,1 satisfy (3) then the the coefficients g" 3,1,2 are
calculated using equation (1).

(d) The coefficients g 3,3, can be chosen freely.

Lemma 2 For N > 5, N / 6,

N-1N- Ricos(n )ja
gd- 3,2,1 0 0 COS ja + 2c
i O j-0

Figure 7: Hermite data merged to define polynomial patch b: (left) Second or-
der Hermite data defining a degree (5,5) patch, (right) Averaged third order data
defining a degree (6,6) patch.

is the unique solution of the system (3).

Proof We apply the discrete Fourier transform (DFT), solve and apply inverse
For N 3 the coefficients g j,k, i > d 3, are a part of a single cubic polyno-
mial, in degree-raised form. For N 4 there are two additional free coefficients
that can be chosen as g_ -3,2, g_3,2,1 and from this an explicit solution can be
constructed. For N 6 there is one additional free coefficient that can be chosen
as gd-3,2,1 and from this an explicit solution can be constructed.
The interplay between the coefficients outside the 3-ring around 0, namely
between g'jk, i < d 3, is simple: the coefficients g _io, g 'd--jj, 2..d-
S- 2, are free and the coefficients g,d-i-, i,d-i-,1 are derived by solving the
two equations (1) and (2).

5 Pointwise Hermite sampling

This section reviews a simple procedure for creating a quadrilateral Bezier patch
b that matches a map f : [0..1]2 Rd up to 2nd order at the covers of [0..1]2.
For degree (5,5) patch, we construct

b: H(f)

-2 4 2 -1
3 3 3 3
4 00 000 OO
-1 7 1 -1
b) 36 36 6
b)* o -- * -- -- -- -
-1 1 -1
41 t 24
O @0 00 06
-1 3 3 -1
4 4 44
d) -* ----
(3,3) (4,3)

Figure 8: Hermite data merged to define macropatches of degree (ij) and smooth-
ness Ci-1,j-1. Formulas a) through d) define unknown Bezier coefficients (hollow
circles) in terms of the coefficients (solid black disks) defined by the H data. For-
mulas a) and b) define coefficients ofa C2 cubic spline. Formulas c) and d) define
coefficients of a C3 spline of degree 4.

by sampling, at each of the four covers of [0..1]2, the partial derivatives up to
second order
f af a f
atf aatf a,2atf
O2f aOtf 2

and placing the Bezier coefficients that define the partial derivatives into one quad-
rant of an array as illustrated in Figures 7. To construct, b := H(f) of degree
(6,6), we average the 3rd derivatives in overlapping positions as illustrated in Fig-
ure 7, right. We can reduce the degree of the sampled patch by choosing it as a
spline rather than as a single polynomial. Figure 8 shows several options. Posi-
tion, first and second derivative at the ends, define a unique C2 spline consisting
of three cubic segments. The formulas for the unknown Bezier points are given
in a) and b). Tensoring this procedure yields a Hermite interpolant consisting of
nine C2 connected bicubic patches as shown in Figure 8 (3,3). Similarly, we can
construct a C3 spline consisting of two degree 4 pieces by the formulas c) and d),
and combine the univariate to tensored bivariate constructions.

6 Guided patchworks

In this section, the sampled C2 segments x', n = 0,..., N 1 are joined to form
a C2 (sprocket layout) or G2 (polar layout) surface rings in R3. The rings, in turn,

are joined into a guided patchwork. First, we consider polar case which is simpler
than the sprocket layout. We set x, := H(g o p).

Lemma 3 For p = the segments x, and x m join C1 and G2.

Proof by Lemma 1 p is C1 and hence

s(g o P, )((0, t) = ,(g o p)(1, t) .
Since k2 is a constant (hence independent of the transversal direction s),

92(gop, )(0,t)= 92(gop)(l,t)+k t gop))l,t)
i.e. circularly adjacent segments of g o p are C1 and G2 connected. Neighboring
patches x' and x match the expansion of H(g o p) at the two endpoints and
the transversal expansions have the same structure. Since k2 is a constant (hence
independent of t),
,x (0, t) ,X (1, t) ,
1x ,(0, t) 92X (t, t) + k20,x,, t)
and the claim follows. 111
Since adjacent reparametrizations "p, p+1 join C2, similar arguments show
that adjacent rings x" and xm+l are also G2 connected.

The sprocket layout construction, based on p = p, samples the guide sur-
face at the corners of the three elementary patches that make up each L-shaped
segment. We explain this as a two-step procedure although it is implemented as
a single sampling. With x" : U H(g o p") for a degree (6, 6) or a degree
(5, 5) patchwork arguments as in Lemma 3 show that adjacent L-shapes are C2
connected (and are internally C2). But, although actual difference is very small,
adjacent rings x- 1 and x" are not even everywhere connected (see Figure 9 left).
The final x" is therefore obtained from x" by replacing the three outermost lay-
ers of Bezier coefficients by a C2 extension of the patch x"-1 (once subdivided to
match the granularity), as shown in Figure 9 middle, right. For m = 0, boundary
data, if any, are extended.
We summarize.
Lemma 4 For p = any finite union of segments and rings UmeN,neZ X (a
patchwork) is a C2 surface. For p = the union is curvature continuous.
The (3, 3) patchwork is constructed analogously, inheriting C2-extended data
from an outer ring before forming the 3 x 3 macro-patch.

Figure 9: Three steps of C2 joining the p-sampled patches of a segment.

7 Guided subdivision surfaces

Joining an infinite sequence of spline rings results in a surface with subdivision
structure (e.g. [RP05, ZoOO, Pra98]. Since their construction and, in the polar
case, even the layout differs from conventional subdivision, we call these surfaces
guided subdivision surfaces. In particular, the first step of the subdivision process,
constructing a guide, sets guided subdivision apart from the usual procedure of
refining meshes. But after the first step, the subdivision process is stationary in the
sense that the refinement rules in terms of the control points of the guide surface
g do not change with each step.

Observation 1 Guided subdivision based on the reparametrizations p = or
p = p is stationary in terms of the controlpoints of the guide surface g and it is a
numerically stable procedure.

Proof Computing the restriction of g, to AA involves only convex combinations,
hence is numerically stable. Since gop' = (gAm)p0, the expression for H(gop')
and hence of the sampled patches x" in terms of the control points of triangular
Bezier patches gnA is independent of the subdivision step. III
To analyze the surface in the limit, we choose the coordinate system so that
x" g(0, 0) (0, 0, 0). Denote the homogeneous part of degree k with respect
to the parameters (u, v) of gn by gn;k. Then gn;kAk Akg;k and x.;k H(g,;k o
Ap0) kH(g,;k o pO) and we obtain the homogeneous decomposition

n A mkXO (4)

Expression (4) immediately implies the following.

Theorem 1 The limit point m -+0 o of x" exists and coincides n i/h the center
point x" of the guide surface.

Now we turn to the more delicate task of showing that guided subdivision surfaces
are C1 manifolds at x' (if the guide surface is such a manifold) and that some of
them (namely degree (6, 6) sprocket and degree (4, 3) polar) are even curvature
continuous. It is tempting to use the curvilinear coordinate system defined by p
and expand g in terms of p to show agreement of the quadratic expansion of x
with the quadratic expansion of g at xC. However, the reparametrizations p are,
by design, singular at the origin and therefore not admissible. In the constructions
[Pra97, Rei98, YZ04], this singularity does not matter, since all points in the
domain are mapped onto the guide surface. Due to sampling, we have to be more
careful, but can still use a similar argument. The following lemma states that
application of H not only results in approximation but also in reproduction of the
lower expansion of the composition.

Lemma 5 For the patchworks defined in Section 6, (i) x', = gn,i o pm. For
degree (6, 6) sprocket and degree (4, 3) polar patchworks x, additionally (ii)
xn;1 + Xn ;2 (gn,1 + gn,2) o pm.
Proof By definition, degree gn,k = k, and the degree of p" is either (3, 3) or
(2, 1). Therefore,
(i) degree(g,,i o p-) degree(pm) < degree(xm1).
(ii) degree((g,,i + gn,2) o p") 2 degree(pm) < degree(xm, + x"2). II
That is, guided subdivision reproduces compositions with linear and quadratic
guide surfaces. We denote by

T>k(S1, ... Si;tl, ... ,tj)

(piecewise) polynomial terms of degree greater than k in each of the variables
si,... Si and with the coefficients depending on tl,... tj. If the coefficients are
constant, we write r>k (1 ,..., Si).
For the remainder, we assume that the tangent planes of g are well-defined in
the vicinity of (0, 0).

Theorem 2 For the patchworks x defined in Section 6, a unique limit of the tan-
gent planes of the surface rings x" exists as m --- o and equals the tangent
plane of the guide surface at x'.

Proof We define n" := cxm x ctxm and note that gn;k depends on n only for
k > 2, i.e. we can define n" below independent of n. From (4) and Lemma 5 (i),

nm(s, t) = A2m(s(g.;1 o0 p) x Ot(g.; o p))(s, ) + Ar>o(s, t; A) r > 3.

n" 0,s(g.;i o po) x t(g.;1 o po)+ .(r-' 7>o(s, t; A)
Inll Il0,g.(giopo) x t(.; o po) +(r.-' T>o(s, t; A)
so that limm ,o i- is normal to the tangent plane of g at x'. I
We may now assume that the coordinate system (x, y, z) in R3 is chosen so
x" = (0, 0, 0) and the tangent plane at x' is {z = 0}. (5)
Since p is injective (see e.g. [PR98]) and the injectivity of p follows directly from
its definition, the projection x -- (x, y) is locally injective near x". Then standard
arguments based on the mean value theorem show that the inverse map is C1 at
the origin (cf [RP05] p.105 (4)). Hence the following theorem holds.

Theorem 3 The guided subdivision surface x is C1 at x' and the tangent plane
of x at x' coincides i/ ith that of guide surface.

The rest of this Section is devoted to the proof of curvature continuity for degree
(6, 6) sprocket and degree (4, 3) polar guided subdivision surfaces.

Definition 1 (curvature continuity) Let f : R2 -+ R3 be tangent plane continu-
ous at fo := f(0, 0). We choose a coordinate system (5) so that the tangent plane
at fo (0, 0, 0) is {z 0 }. Assume further that for every (s, t) / (0, 0) there
exists a unique (elliptic or hyperbolic) osculating paraboloid q := (qo,..., q5)T
anchored at (s, t), i.e. i/ ith the qi continuous functions of(s, t) such that

for (x,y, z) f(s + s, t + t), z = (x2, xy, 2, y, 1)q(s, t) + >2(S, t; s,t).
Then the surface f is curvature continuous at fo ifthe anchored osculating paraboloids
have a unique limit at fo.

The advantage of fixing the coordinate system of the osculating paraboloid at
(s, t) over defining it at (s + s, t + t) as is common in differential geometry is
that the anchored osculating paraboloid does not vary if data are sampled from the
paraboloid (see Figure 10).

zi ..o /

(xo, zo)

Figure 10: The osculating parabolas (dashed) vary with the local coordinate sys-
tem at points xioc, zloc on an anchored osculating parabola (solid).

Theorem 4 (curvature continuity) sprocket degree (6, 6) andpolar degree (4, 3)
guided subdivision surfaces x are curvature continuous at the extraordinary point
x" if the guide surface g has a unique osculating paraboloid q at xC. The limit of
the osculating paraboloids of x at x' is q.

Proof We fix a coordinate system (5) so that, near x",

g = (u + 7r>i(u, v), v + 7r>,(u, v), au2 + buv + cv2 + 7>(U, v))

With (s + s, t + t a point near (s, t) E E, we expand po to second degree at (s, t)

po =(u(s + S, t + ), v(s + s, t + ))
=(u(s, t; s, t + 7>2(S, t; s, t), v(s, t; s, t) + >2(s, t; s, t))
u(s, t; s, t) :uo + hs + h2+ h42 + h6t+ h8t,
v(s, t; s, t) :=vo + his + h3t + h5s2 + h7t + h9t2 .

Note that Uo, i,, and hi are piecewise polynomials in s and t. Since po is injective
on the compact domain E x ZN, hoh3 hah2 > const > 0. By the homogeneous
expansion (4),
XO Z l X;k, /- A ,

, Xloc

and by Lemma 5 (ii),

x"' = I(g.; 1 PO) + 1g;2 O ) 13
X, cnl /nI n0p 2(gn;,2 Pon)+1Y(...)
S(lu + 12El, v + 12E2, 12(a2 + bu + cv2) + 13E3) + 7>2(, ; S, t, l),
where Ek are some polynomials of total degree 2 in (s, t) with coefficients de-
pending on s, t, 1. This expression for x (x, y, z) is substituted into the
anchored osculating paraboloids of x. Near x", with q := (a, b, c, 0, 0, 0)T con-
stant, independent of (s, t) and d := (do,..., d5)T, each term depending on (s, t),
the osculating paraboloid has the form

z = (2, xy,y2, ,y, 1) (q + d). (8)

Equating the coefficients of 1, s, t, 2, st, t2 to 0, we get the system of six linear
equations in the six unknowns dj(s, t),

Md = r (9)

where the first three columns of M have a factor 12, the next two a factor I and M is
of full rank apart from (0, 0) since detM -1((hoh3 h1h2)4 + 7>s(l; s, t) / 0.
Since the right hand side r has a factor 13, hence vanishes faster than the matrix
entries, limo dj = 0. This satisfies Definition 1. 111
The lower degree bound for curvature continuous subdivision surfaces [Rei96]
applies to guided subdivision surfaces with sprocket layout; the curvature contin-
uous polar surfaces of degree (4, 3) do not contradict this bound since the patch
layout is different from the one assumed in [Rei96]. The layout allows reducing
the radial degree to 3 and replacing C2 continuity by geometric G2 continuity
allows reducing the circular degree to 4.
To see that curvature continuous guided subdivision surfaces are C2 from
the point of view of differential geometry, we consider a local parametrization
(x, y, h(x, y)) of the surface at xC. The height function h is C1 by Theorem 3 and
the partial derivatives hx, hy are C1 apart from (0, 0), by construction (and since
the limit osculating paraboloid q is anchored at x' and therefore, in particular, the
constant component of q vanishes faster than the other qi). The G2 connection for
polar layout does not create a problem since the G2 constraints reparametrize a
C2 join. Theorem 4 shows that first partial derivatives of hx, hy are well defined
as lim(X,y)-(o,o). The mean value theorem implies that hx, hy are C1 at the origin.
Hence h is C2 at the origin.


.....: .. 3.7

Figure 11: Gauss curvature shading. (top) sprocket layout subdivision of degree
(6,6) (left), (5,5) (middle) and (3,3) (right). (bottom) Polar layout subdivision of
degree (4,3) (left), and (3,3) (middle). (bottom,right) juxtaposes the sprocket (top)
and polar (bottom) C2 surfaces.

8 Curvature estimates for lower-degree guided sub-

Here we bound the curvature at the extraordinary point when the degree of x is
chosen lower than (6, 6) in the sprocket case and (4, 3) in the polar case. The
trade-off between quality and degree will be discussed in more detail in a later
report although Figure 11 gives a first impression. Lower degree is often required
by an application and we will give an algorithm for practically computing the
bounds. Also, the discussion highlights the computational value of introducing
anchored osculating paraboloids.

Theorem 5 Guided subdivision surfaces of degree (3,3) and higher have bounded
curvature at the extraordinary point.

Proof We follow the proof of Theorem 4 but replace formula (7) by
x =(l + 12EI, l + 12E2, + +7>2(S, t; S, t, 1) ,
v E2, 13E3)+ (10)
z :=ho+ hO s + h2t + 3s2 + h4St + h2

and the paraboloid by z = (x2, y,, x, y, 1)d since Lemma 5 (ii) may not hold
and no osculating paraboloid exists anchored at fo whose perturbation we would

consider. We obtain a constraint matrix M of full rank and with the same factors
of I as (9). However, the right hand side r now has a factor of 12, enough only to
lim d = for j = 3, 4, 5.
We will compute bounds on do, dl, d2 by applying Cramer's rule. Since the factor
12 appears both in r and the relevant first three columns of M, we can remove the
factors from the matrix and the right hand and write them respectively as

2 +. uovo+. v2+. uo+. vo+. 1 ho+.
2houo+. hovo+hluo+. 2hivo+. ho+. hi+. 0 il+
2h2uo+. h2vo+h3uo+. 2h3vo+. h2+. h3+. 0 h2
h +2h40o+. hoh0+h4vo+h5uo+. h2+2h5vo+. h4+. h5+. 0 3+
2hoh2+2h6uo+. hoh3+hlh2+h6vo+h7uo+. 2hih3+2h7vo h+. h7+. 0. 4 +.
h~+2hsuo+. h2h3+hsvo+hguo+. h2+2hgvo+. hs+. h9+.+ hs+.

where +. is a shorthand for polynomial terms 7r>o(l; s, t) that will vanish as 1 0.
Elimination of the first row and multilinearity of determinants yields a simpler

0+. 0+. 0+. ho+. hi+ d h
0+. 0+. 0+. h2+. h3+. d h2+
h2+. hohi+. h2+. h4+. hs+. d2 3 +. (12)
2hoh2+. hoh3+hlh2+. 2hlh3+. h6+. h7+. 4 h4
h|+. h2h3+. h+. h8+. h9+. d5 h5+.

Since hoh3 hh2 is bounded away from zero due to the injectivity of p, detM
(hoh3 hh2)4 +. > cost > 0 and D := detlimlo M is well-defined. Let Di
be the determinant of the matrix obtained by replacing the (i + 1)th column of M
by r and by taking the limit for 1 -- 0 to drop the terms +., we have by Cramer's
di ,i 0,1,2.
Since the denominator is well-defined, the di are bounded. III
The practical calculation of the bounds on the coefficients do, dl and d2 of
the osculating paraboloid is simplified since D and Di share factors hoh3 hlh2.
Specifically, we compute as follows.
(i) Bounding the coefficients do, di, d2 for functions fl := u2, f2 : uv, f3 := 2
in turn yields nine intervals that allow computing the bounds for any function
f 712 2 + 72UU ',
(ii) If (x, y, z) is an orthogonal coordinate system in R3 with xC = (0, 0, 0), tan-
gent plane {z = 0} at xc and g = (elu + el2v + 7r>i(u, v), e2u + e22v +
7r>i(u, v), au2 + buv + cv2 + 7r>2(U, v)), then the bounds of coefficients d' of

osculating paraboloid in such system are calculated from the previous bounds of
coefficients dk as
d'o doe2 dieoel + d2e d doe + dl2e3 + d2
d1 =2doeoe2 + dl(ee3 + ele2) + 2d2e613 ,

where ( 02) : (11 e12)-'
where l e3 e \ 21 e22
(iii) To bound Gaussian and mean curvatures, we observe that for an osculating
paraboloid (x, y, d' Z2 + d'xy + d'y2) the mean H and Gaussian K curvatures at
the origin (0, 0, 0) are
H = do + d K = 4dd' d .2

substituting formulas (13), we get
H (e2 + e2)2do + (eoeI + e623)dl + (e2 + e)d2 ,
K =(eoe3 ei2)2(4dod2 d) .
For Gaussian curvature K, we get a tighter estimate if, in formula (14), the part
4dod2 d1 is precomputed with respect basis functions fl, f2, f3. The result are,
consistent with [PetO1], six precomputed intervals and nine intervals for H.
(iv) Defining fl := u2 + v2, f2 : uv, f3 := v2 gives an immediate impression of
how guided subdivision reproduces canonical elliptic, hyperbolic and parabolic
shapes. We list some bounds on the Gaussian curvature for these shapes when
N 8.
(3,3) C (5, 5) C (3,3)Pol
u2 + 2 [3.87501,4.22213] [3.96505, 4.02945] [3.98505, 4.03352]
uv [-1.06493,-0.96478] [-1.00426,-0.99261] [-1.05229,-0.97047]
v2 [-0.15738, 0.15644] [-0.02054, 0.02241] [-0.09641, 0.05184]

9 The accelerated bicubic guided C2 scheme

We consider a sampling scheme that is no longer stationary, since, in each step,
we sample with increasing density. Each quad in SO is evenly subdivided into 4m
subquads at level m and H is applied over each subquad creating 4" of pieces of
3 x 3 bicubic patches that are joined C2 as explained in Section 5. We call this
scheme accelerated bicubic and observe that the sampled data will be sufficiently
dense after a couple of steps to meet the bounds needed for the proof below so
that we can switch to a fixed density, pseudostationary sampling.

Theorem 6 Accelerated bicubic guided subdivision surfaces x are curvature con-
tinuous at xC. Moreover, the limit of osculating paraboloids of x at xc coincide
i ith that of guide surface g.

Proof The proof follows that of Theorem 5. By Theorem 2, accelerated subdi-
vision inherits the tangent plane of the guide surface at xc. To show curvature
continuity, we observe that if a polynomial g o p : [0..1]2 IR is bicubically
sampled at the covers (s, t) of 4" subquads to define a segment x" consisting of
4m patches then, for a fixed c > 0 and m sufficiently large, the value and partial
derivatives of g o p and x' differ everywhere by less than c. Since the domain is
compact, also the functions hi, hj in (12) deviate from their value at the corners
by less than a fixed c and all expressions (+.) are bounded and converge to zero
with m. With q := (a, b, c 0, 0)T the osculating paraboloid of g at (0, 0), di
converges to qi: limlo do a limo di b limlo d2 c. III
As in Section 7, one can argue that the accelerated subdivision surfaces are C2
in the sense of differential geometry.

10 Rational guided subdivision

If the guide surface is rational, e.g. g : (fl/f4, f2/f4, f3/f4), we can either sam-
ple in R3 (and all proofs and theorems apply to the resulting polynomial guided
subdivision surface) or, we sample the homogeneous guide (fl, f2, f3, f4) in R4
and project the result to R3. The latter allows reproducing, e.g. the sphere, exactly.
As a stationary construction, this differs from the approach in [MWW01] (which
is akin to accelerated scheme in that the density increases at the poles) and from
[SZSS98, SZBN03].

11 Conclusions

This paper defined local subdivision constructions that, for practical applications,
would be embedded in a larger scheme that separates local subdivision regions by
constructing and pairwise blending primary surfaces. Such schemes, for example
one or more steps of Catmull-Clark subdivision, are discussed in [KPOx]. [KPOx]
also explains how to choose the free parameters of the guide surface g to transi-
tion from primary surfaces to the local subdivision surfaces without introducing
unnecessary shape artifacts, and how to stop the subdivision process to fill in the
remaining multi-sided hole with polynomial pieces of moderate degree.

Here we established that any finite number of surface rings join to form a C2
surface (2-manifold with boundary) with a gap at the center; and, in the limit,
the construction yields curvature continuous subdivision surfaces of low degree.
A generalization to higher-order smoothness looks straightforward although the
details, e.g. the choice of reparametrizations, requires attention.
Acknowledgement: This work was supported by NSF Grant CCF-0430891.


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